The Perfect-Graph Robust Algorithm Problem

An algorithm, which for any easily recognizable input, A, finds either an
easily recognizable B or an easily recognizable C, is sometimes called a
"robust algorithm" - to be distinguished from a non-robust algorithm,
which, for any A without a B, finds a C. Either provides a proof of the
"existentially polytime (EP) theorem": For any A, there exists a B or a
C. In other words: For any A without a B, there is a C.

In [
MR 92i:68043], Jack Edmonds and Kathie Cameron advocated seeking a robust
algorithm which, for any graph G, finds either a clique and colouring the
same size or else finds an easily recognizable combinatorial obstruction
to G being perfect. The obstruction might be specified to be a
"alpha-omega partitioned subgraph", or it might be specified more
particularly to be an odd hole or odd antihole.

Such an algorithm might be simpler than an algorithm for recognizing
whether or not a graph is perfect, in view of precedents, and since what
it would do is incomparable with perfect-graph recognition. Such an
algorithm could end up giving a clique and colouring the same size in a
non-perfect graph.

Here are two examples of similar problems which have been solved.
Edmonds has given a simple robust algorithm which, for any graph G,
either finds an odd cycle with at most one chord (a defining
obstruction to G being Meyniel) or else finds a clique and colouring the
same size. This is an improvement on the non-robust algorithms of Hoang
and Hertz which, assuming a graph is Meyniel, find a clique and colouring
the same size. Edmonds' algorithm is much simpler than the
Burlet-Fonlupt decomposition algorithm for recognizing Meyniel graphs,
which was motivated by an interest in optimizing in Meyniel graphs, and
which is used by Hoang and Hertz.

Conforti and Cornuejols give a complicated decomposition algorithm for
recognizing whether or not a matrix is balanced. At about the
same time, to motivate the advocacy of a robust algorithm for either
node-colouring a graph or recognizing it to be not perfect, Cameron and
Edmonds presented a simple algorithm which, for any 0-1 matrix M, either
finds, where x is the largest number of ones in any row, an x-colouring
of the columns so that the 1's of any row are in different coloured
columns, or else finds "an odd hole" in M (the defining obstruction to M
being balanced). This introduced the "EP - robust algorithm" paradigm
which is followed in Edmonds' Meyniel-related algorithm, and is related
to the Conforti-Cornuejols-Rao treatment of balanced matrices in the same
way that Edmonds' is related to the Burlet-Fonlupt treatment of Meyniel
graphs. Following the same paradigm we expect there to be a robust
algorithm proving the SPCG, related in the same way to the
Chudnovsky-Robertson-Seymour-Thomas decomposition of Berge graphs.

In conclusion, we know the following

EP Theorem 1: For any graph, there is either a clique and a colouring of
the same size, or there is an alpha-omega partitioned subgraph (or both).

EP Theorem 2 (SPGT): For any graph, there is either a clique and a
colouring of the same size, or there is a odd hole or odd antihole (or
both). So:
Give a combinatorial polytime algorithm to find what the EP theorem
asserts to exist.